-6n^2-n+6=0

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Solution for -6n^2-n+6=0 equation:


Simplifying
-6n2 + -1n + 6 = 0

Reorder the terms:
6 + -1n + -6n2 = 0

Solving
6 + -1n + -6n2 = 0

Solving for variable 'n'.

Begin completing the square.  Divide all terms by
-6 the coefficient of the squared term: 

Divide each side by '-6'.
-1 + 0.1666666667n + n2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 0.1666666667n + 1 + n2 = 0 + 1

Reorder the terms:
-1 + 1 + 0.1666666667n + n2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 0.1666666667n + n2 = 0 + 1
0.1666666667n + n2 = 0 + 1

Combine like terms: 0 + 1 = 1
0.1666666667n + n2 = 1

The n term is 0.1666666667n.  Take half its coefficient (0.08333333335).
Square it (0.006944444447) and add it to both sides.

Add '0.006944444447' to each side of the equation.
0.1666666667n + 0.006944444447 + n2 = 1 + 0.006944444447

Reorder the terms:
0.006944444447 + 0.1666666667n + n2 = 1 + 0.006944444447

Combine like terms: 1 + 0.006944444447 = 1.006944444447
0.006944444447 + 0.1666666667n + n2 = 1.006944444447

Factor a perfect square on the left side:
(n + 0.08333333335)(n + 0.08333333335) = 1.006944444447

Calculate the square root of the right side: 1.003466215

Break this problem into two subproblems by setting 
(n + 0.08333333335) equal to 1.003466215 and -1.003466215.

Subproblem 1

n + 0.08333333335 = 1.003466215 Simplifying n + 0.08333333335 = 1.003466215 Reorder the terms: 0.08333333335 + n = 1.003466215 Solving 0.08333333335 + n = 1.003466215 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.08333333335' to each side of the equation. 0.08333333335 + -0.08333333335 + n = 1.003466215 + -0.08333333335 Combine like terms: 0.08333333335 + -0.08333333335 = 0.00000000000 0.00000000000 + n = 1.003466215 + -0.08333333335 n = 1.003466215 + -0.08333333335 Combine like terms: 1.003466215 + -0.08333333335 = 0.92013288165 n = 0.92013288165 Simplifying n = 0.92013288165

Subproblem 2

n + 0.08333333335 = -1.003466215 Simplifying n + 0.08333333335 = -1.003466215 Reorder the terms: 0.08333333335 + n = -1.003466215 Solving 0.08333333335 + n = -1.003466215 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.08333333335' to each side of the equation. 0.08333333335 + -0.08333333335 + n = -1.003466215 + -0.08333333335 Combine like terms: 0.08333333335 + -0.08333333335 = 0.00000000000 0.00000000000 + n = -1.003466215 + -0.08333333335 n = -1.003466215 + -0.08333333335 Combine like terms: -1.003466215 + -0.08333333335 = -1.08679954835 n = -1.08679954835 Simplifying n = -1.08679954835

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.92013288165, -1.08679954835}

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